Partitions with "d(a) copies of a"

In [1], Agarwal & Andrews studied partitions with "a copies of a," and in [2], Agarwal & Mullen studied partitions with "d(a) copies of a" (where d is the divisor function). In this note, partitions with "M(a) copies of a" are considered; the maximum exponent function, M, is defined by M(a) = max(e l5. .. , e T) if the integer a > 1 has canonical prime-power form a = p^ 1 ... p &r; , and Af(l) = 1. Define L to be the set of ordered pairs (a, b) of positive integers with 1 < b < M{a). We say i is a partition of n with M{a) copies of a if IT is a finite ordered collection (a l5 b\) , (di* b^_) , ..., (a^, b k) of elements of L such that GL\ + CLi + • • • + a^ = n and, for I < i < j < k, a^ > a-j with bi < bj if CLI = 0 ^e# The factor (1-q n)~ • •-(1 + q nMin) + ^ n ^) +n «(«) + ...) for rii = n (1 < i < Af(n)) ; thus, the number of partitions of n with "M(a) copies of a" is counted. For example, TT?(4) is the coefficient of q^ in then w(4) = 6, and the exponents 298 [Nov.